56.4.62 problem 59

Internal problem ID [8951]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 59
Date solved : Monday, January 27, 2025 at 05:20:41 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }+y^{\prime }+y&=x \,{\mathrm e}^{x} \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 69

Order:=6; 
dsolve((1-x^2)*diff(y(x),x$2)+diff(y(x),x)+y(x)=x*exp(x),y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {7}{120} x^{5}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) y^{\prime }\left (0\right )+\frac {x^{3}}{6}+\frac {x^{4}}{24}+\frac {7 x^{5}}{120}+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 63

AsymptoticDSolveValue[(1-x^2)*D[y[x],{x,2}]+D[y[x],x]+y[x]==x*Exp[x],y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {x^5}{120}-\frac {x^4}{24}-\frac {x^2}{2}+x\right )+c_1 \left (\frac {7 x^5}{120}-\frac {x^4}{12}+\frac {x^3}{6}-\frac {x^2}{2}+1\right ) \]